hypothesis testing statistics

Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

Hypothesis Testing

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis ($H_{0} $) and the alternative hypothesis ($H_{1}$) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark ($\mu$).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

$H_{0}$: $\mu = 170 \: cm $

$H_{1}$: $\mu > 170 \: cm $

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as ($H_{A} $).

The Significance Level

The significance level ($\alpha$) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

$\alpha = 0.1$ (10%)
$\alpha = 0.05$ (5%)
$\alpha = 0.01$ (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

Standard Normal Distribution (Z): used for Testing Population Proportions
Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

The critical value approach compares the test statistic with the critical value of the significance level.
The p-value approach compares the p-value of the test statistic and with the significance level.

The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level ($\alpha$).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level ($\alpha = 0.05$):

We reject the null hypothesis ($H_{0} $) at 0.05 significance level ($\alpha$)

The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level ($\alpha$).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis ($H_{0} $) at a 0.05 significance level ($\alpha$)

We keep the null hypothesis ($H_{0}$) at a 0.01 significance level ($\alpha$)

Note: The two approaches are only different in how they present the conclusion.

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

Check the conditions
Define the claims
Decide the significance level
Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

Proportions (for qualitative data)
Mean values (for numerical data)

You will learn the steps for both types in the following pages.

COLOR PICKER

Contact Sales

If you want to use W3Schools services as an educational institution, team or enterprise, send us an e-mail: [email protected]

Report Error

If you want to report an error, or if you want to make a suggestion, send us an e-mail: [email protected]

Hypothesis Testing

Already have an account? Log in here.

A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the $p$-value to the level of significance (a chosen target). If the $p$-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis $($denoted $H_0)$ is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis $($denoted $H_1).$ Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted $\alpha$, which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted $\beta$, which is also its probability of occurrence. Also, $\alpha$ is known as the significance level , and $1-\beta$ is known as the power of the test. $H_0$ $\textbf{is true}$$\hspace{15mm}$ $H_0$ $\textbf{is false}$ $\textbf{Reject}$ $H_0$$\hspace{10mm}$ Type I error Correct Decision $\textbf{Reject}$ $H_1$ Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The $p$-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator $\hat \theta$ of a population statistic $\theta$, following a probability distribution $P(T)$, computed from a sample $\mathcal{S},$ and given a significance level $\alpha$ and test statistic $t^*,$ define $H_0$ and $H_1;$ compute the test statistic $t^*.$ $p$-value Approach (most prevalent): Find the $p$-value using $t^*$ (right-tailed). If the $p$-value is at most $\alpha,$ reject $H_0$. Otherwise, reject $H_1$. Critical Value Approach: Find the critical value solving the equation $P(T\geq t_\alpha)=\alpha$ (right-tailed). If $t^*>t_\alpha$, reject $H_0$. Otherwise, reject $H_1$. Note: Failing to reject $H_0$ only means inability to accept $H_1$, and it does not mean to accept $H_0$.

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the $p$-value approach with significance level $\alpha=0.05:$ Define $H_0$: $\mu=200$. Define $H_1$: $\mu>200$. Since our values are normally distributed, the test statistic is $z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09$. Using a standard normal distribution, we find that our $p$-value is approximately $0.001$. Since the $p$-value is at most $\alpha=0.05,$ we reject $H_0$. Therefore, we can conclude that the test shows sufficient evidence to support the claim that $\mu$ is larger than $200$ mg/dL.

If the sample size was smaller, the normal and $t$-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the $p$-value approach with significance level $\alpha=0.05$ and the $t$-distribution with 24 degrees of freedom: Define $H_0$: $\mu=200$. Define $H_1$: $\mu\neq 200$. Using the $t$-distribution, the test statistic is $t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54$. Using a $t$-distribution with 24 degrees of freedom, we find that our $p$-value is approximately $2(0.068)=0.136$. We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the $p$-value is larger than $\alpha=0.05,$ we fail to reject $H_0$. Therefore, the test does not show sufficient evidence to support the claim that $\mu$ is not equal to $200$ mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level $\alpha$) for a population parameter $\theta$ is equivalent to finding a confidence interval $($with confidence level $1-\alpha)$ for the population parameter $\theta$. If the assumption on the parameter $\theta$ falls inside the confidence interval, then the test has failed to reject the null hypothesis $($with $p$-value greater than $\alpha).$ Otherwise, if $\theta$ does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate $($with $p$-value at most $\alpha).$

Statistics (Estimation)
Normal Distribution
Correlation
Confidence Intervals

Problem Loading...

Note Loading...

Set Loading...

Hypothesis Testing

About hypothesis testing.

Watch the video for a brief overview of hypothesis testing:

Can’t see the video? Click here to watch it on YouTube.

Contents (Click to skip to the section):

What is a Hypothesis?

What is hypothesis testing.

Hypothesis Testing Examples (One Sample Z Test).
Hypothesis Test on a Mean (TI 83).

Bayesian Hypothesis Testing.

More Hypothesis Testing Articles
Hypothesis Tests in One Picture
Critical Values

What is the Null Hypothesis?

Need help with a homework problem? Check out our tutoring page!

A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example:

A new medicine you think might work.
A way of teaching you think might be better.
A possible location of new species.
A fairer way to administer standardized tests.

It can really be anything at all as long as you can put it to the test.

What is a Hypothesis Statement?

If you are going to propose a hypothesis, it’s customary to write a statement. Your statement will look like this: “If I…(do this to an independent variable )….then (this will happen to the dependent variable ).” For example:

If I (decrease the amount of water given to herbs) then (the herbs will increase in size).
If I (give patients counseling in addition to medication) then (their overall depression scale will decrease).
If I (give exams at noon instead of 7) then (student test scores will improve).
If I (look in this certain location) then (I am more likely to find new species).

A good hypothesis statement should:

Include an “if” and “then” statement (according to the University of California).
Include both the independent and dependent variables.
Be testable by experiment, survey or other scientifically sound technique.
Be based on information in prior research (either yours or someone else’s).
Have design criteria (for engineering or programming projects).

Hypothesis testing can be one of the most confusing aspects for students, mostly because before you can even perform a test, you have to know what your null hypothesis is. Often, those tricky word problems that you are faced with can be difficult to decipher. But it’s easier than you think; all you need to do is:

Figure out your null hypothesis,
State your null hypothesis,
Choose what kind of test you need to perform,
Either support or reject the null hypothesis .

If you trace back the history of science, the null hypothesis is always the accepted fact. Simple examples of null hypotheses that are generally accepted as being true are:

DNA is shaped like a double helix.
There are 8 planets in the solar system (excluding Pluto).
Taking Vioxx can increase your risk of heart problems (a drug now taken off the market).

How do I State the Null Hypothesis?

You won’t be required to actually perform a real experiment or survey in elementary statistics (or even disprove a fact like “Pluto is a planet”!), so you’ll be given word problems from real-life situations. You’ll need to figure out what your hypothesis is from the problem. This can be a little trickier than just figuring out what the accepted fact is. With word problems, you are looking to find a fact that is nullifiable (i.e. something you can reject).

Hypothesis Testing Examples #1: Basic Example

A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks.

The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: H 1 : μ > 8.2

Next, you’ll need to state the null hypothesis . That’s what will happen if the researcher is wrong . In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: H 0 μ ≤ 8.2

Rejecting the null hypothesis

Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy— the hard part is usually figuring out what your null hypothesis is in the first place.

Hypothesis Testing Examples (One Sample Z Test)

The one sample z test isn’t used very often (because we rarely know the actual population standard deviation ). However, it’s a good idea to understand how it works as it’s one of the simplest tests you can perform in hypothesis testing. In English class you got to learn the basics (like grammar and spelling) before you could write a story; think of one sample z tests as the foundation for understanding more complex hypothesis testing. This page contains two hypothesis testing examples for one sample z-tests .

One Sample Hypothesis Testing Example: One Tailed Z Test

Watch the video for an example:

A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.

Step 1: State the Null hypothesis . The accepted fact is that the population mean is 100, so: H 0 : μ = 100.

Step 2: State the Alternate Hypothesis . The claim is that the students have above average IQ scores, so: H 1 : μ > 100. The fact that we are looking for scores “greater than” a certain point means that this is a one-tailed test.

Step 4: State the alpha level . If you aren’t given an alpha level , use 5% (0.05).

Step 5: Find the rejection region area (given by your alpha level above) from the z-table . An area of .05 is equal to a z-score of 1.645.

Step 6: If Step 6 is greater than Step 5, reject the null hypothesis. If it’s less than Step 5, you cannot reject the null hypothesis. In this case, it is more (4.56 > 1.645), so you can reject the null.

One Sample Hypothesis Testing Examples: #3

Watch the video for an example of a two-tailed z-test:

Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect.

State the null hypothesis : H 0 :μ=100
State the alternate hypothesis : H 1 :≠100
State your alpha level. We’ll use 0.05 for this example. As this is a two-tailed test, split the alpha into two. 0.05/2=0.025
Find the z-score associated with your alpha level . You’re looking for the area in one tail only . A z-score for 0.75(1-0.025=0.975) is 1.96. As this is a two-tailed test, you would also be considering the left tail (z = 1.96)
If Step 5 is less than -1.96 or greater than 1.96 (Step 3), reject the null hypothesis . In this case, it is greater, so you can reject the null.

*This process is made much easier if you use a TI-83 or Excel to calculate the z-score (the “critical value”). See:

Critical z value TI 83
Z Score in Excel

Hypothesis Testing Examples: Mean (Using TI 83)

You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses; that’s up to you to read the question and input it into the calculator.

Example problem : A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.

Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0 : μ = 18.9

Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis: H 1 : μ ≠ 18.9

Step 3: Press Stat then press the right arrow twice to select TESTS.

Step 4: Press 1 to select 1:Z-Test… . Press ENTER.

Step 5: Use the right arrow to select Stats .

Step 6: Enter the data from the problem: μ 0 : 18.9 σ: 5 x : 21 n: 200 μ: ≠μ 0

Step 7: Arrow down to Calculate and press ENTER. The calculator shows the p-value: p = 2.87 × 10 -9

This is smaller than our alpha value of .05. That means we should reject the null hypothesis .

Bayesian Hypothesis Testing: What is it?

Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated? Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cell-phone towers are to blame. However, a second study found that cell-phone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen simply by chance . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.

It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a p-value . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.

Traditional testing (the type you probably came across in elementary stats or AP stats) is called Non-Bayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable. Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?

Differences Between Traditional and Bayesian Hypothesis Testing.

Traditional testing (Non Bayesian) requires you to repeat sampling over and over, while Bayesian testing does not. The main different between the two is in the first step of testing: stating a probability model. In Bayesian testing you add prior knowledge to this step. It also requires use of a posterior probability , which is the conditional probability given to a random event after all the evidence is considered.

Arguments for Bayesian Testing.

Many researchers think that it is a better alternative to traditional testing, because it:

Includes prior knowledge about the data.
Takes into account personal beliefs about the results.

Arguments against.

Including prior data or knowledge isn’t justifiable.
It is difficult to calculate compared to non-Bayesian testing.

Hypothesis Testing Articles

What is Ad Hoc Testing?
Composite Hypothesis Test
What is a Rejection Region?
What is a Two Tailed Test?
How to Decide if a Hypothesis Test is a One Tailed Test or a Two Tailed Test.
How to Decide if a Hypothesis is a Left Tailed Test or a Right-Tailed Test.
How to State the Null Hypothesis in Statistics.
How to Find a Critical Value .
How to Support or Reject a Null Hypothesis.

Specific Tests:

Brunner Munzel Test (Generalized Wilcoxon Test).
Chi Square Test for Normality.
Cochran-Mantel-Haenszel Test.
Granger Causality Test .
Hotelling’s T-Squared.
KPSS Test .
What is a Likelihood-Ratio Test?
Log rank test .
MANCOVA Assumptions.
MANCOVA Sample Size.
Marascuilo Procedure
Rao’s Spacing Test
Rayleigh test of uniformity.
Sequential Probability Ratio Test.
How to Run a Sign Test.
T Test: one sample.
T-Test: Two sample .
Welch’s ANOVA .
Welch’s Test for Unequal Variances .
Z-Test: one sample .
Z Test: Two Proportion.
Wald Test .

What is an Acceptance Region?
How to Calculate Chebyshev’s Theorem.
Contrast Analysis
Decision Rule.
Degrees of Freedom .
Directional Test
False Discovery Rate
How to calculate the Least Significant Difference.
Levels in Statistics.
How to Calculate Margin of Error.
Mean Difference (Difference in Means)
The Multiple Testing Problem .
What is the Neyman-Pearson Lemma?
What is an Omnibus Test?
One Sample Median Test .
How to Find a Sample Size (General Instructions).
Sig 2(Tailed) meaning in results
What is a Standardized Test Statistic?
How to Find Standard Error
Standardized values: Example.
How to Calculate a T-Score.
T-Score Vs. a Z.Score.
Testing a Single Mean.
Unequal Sample Sizes.
Uniformly Most Powerful Tests.
How to Calculate a Z-Score.

Tutorial Playlist

Statistics tutorial, everything you need to know about the probability density function in statistics, the best guide to understand central limit theorem, an in-depth guide to measures of central tendency : mean, median and mode, the ultimate guide to understand conditional probability.

A Comprehensive Look at Percentile in Statistics

The Best Guide to Understand Bayes Theorem

Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, chi-square test, what is hypothesis testing in statistics types and examples, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, mean squared error: overview, examples, concepts and more, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

All You Need to Know About Bias in Statistics

A Complete Guide to Get a Grasp of Time Series Analysis

The Key Differences Between Z-Test Vs. T-Test

The Complete Guide to Understand Pearson's Correlation

A complete guide on the types of statistical studies, everything you need to know about poisson distribution, your best guide to understand correlation vs. regression, the most comprehensive guide for beginners on what is correlation, hypothesis testing in statistics - types | examples.

Lesson 10 of 24 By Avijeet Biswal

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis and hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life -

A teacher assumes that 60% of his college's students come from lower-middle-class families.
A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

The Ultimate Ticket to Top Data Science Job Roles

Importance of Hypothesis Testing in Data Analysis

Here is what makes hypothesis testing so important in data analysis and why it is key to making better decisions:

Avoiding Misleading Conclusions (Type I and Type II Errors)

One of the biggest benefits of hypothesis testing is that it helps you avoid jumping to the wrong conclusions. For instance, a Type I error could occur if a company launches a new product thinking it will be a hit, only to find out later that the data misled them. A Type II error might happen when a company overlooks a potentially successful product because their testing wasn’t thorough enough. By setting up the right significance level and carefully calculating the p-value, hypothesis testing minimizes the chances of these errors, leading to more accurate results.

Making Smarter Choices

Hypothesis testing is key to making smarter, evidence-based decisions. Let’s say a city planner wants to determine if building a new park will increase community engagement. By testing the hypothesis using data from similar projects, they can make an informed choice. Similarly, a teacher might use hypothesis testing to see if a new teaching method actually improves student performance. It’s about taking the guesswork out of decisions and relying on solid evidence instead.

Optimizing Business Tactics

In business, hypothesis testing is invaluable for testing new ideas and strategies before fully committing to them. For example, an e-commerce company might want to test whether offering free shipping increases sales. By using hypothesis testing, they can compare sales data from customers who received free shipping offers and those who didn’t. This allows them to base their business decisions on data, not hunches, reducing the risk of costly mistakes.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

Here, x̅ is the sample mean,
μ0 is the population mean,
σ is the standard deviation,
n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

Your Dream Career is Just Around The Corner!

Null Hypothesis and Alternative Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average.

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

Become a Data Scientist with Hands-on Training!

Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. To infer conclusions accurately, this data should be representative of the population.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

3. Chi-Square

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

ANOVA , or Analysis of Variance, is a statistical method used to compare the means of three or more groups. It’s particularly useful when you want to see if there are significant differences between multiple groups. For instance, in business, a company might use ANOVA to analyze whether three different stores are performing differently in terms of sales. It’s also widely used in fields like medical research and social sciences, where comparing group differences can provide valuable insights.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

Become a Data Scientist through hands-on learning with hackathons, masterclasses, webinars, and Ask-Me-Anything sessions! Start learning!

Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

Become a Data Scientist With Real-World Experience

Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

The null hypothesis is (H0 <= 90) or less change.
A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Serious About Success? Don't Settle for Less

Practice Problems on Hypothesis Testing

Here are the practice problems on hypothesis testing that will help you understand how to apply these concepts in real-world scenarios:

A telecom service provider claims that customers spend an average of ₹400 per month, with a standard deviation of ₹25. However, a random sample of 50 customer bills shows a mean of ₹250 and a standard deviation of ₹15. Does this sample data support the service provider’s claim?

Solution: Let’s break this down:

Null Hypothesis (H0): The average amount spent per month is ₹400.
Alternate Hypothesis (H1): The average amount spent per month is not ₹400.
Population Standard Deviation (σ): ₹25
Sample Size (n): 50
Sample Mean (x̄): ₹250

1. Calculate the z-value:

z=250-40025/50 −42.42

2. Compare with critical z-values: For a 5% significance level, critical z-values are -1.96 and +1.96. Since -42.42 is far outside this range, we reject the null hypothesis. The sample data suggests that the average amount spent is significantly different from ₹400.

Out of 850 customers, 400 made online grocery purchases. Can we conclude that more than 50% of customers are moving towards online grocery shopping?

Solution: Here’s how to approach it:

Proportion of customers who shopped online (p): 400 / 850 = 0.47
Null Hypothesis (H0): The proportion of online shoppers is 50% or more.
Alternate Hypothesis (H1): The proportion of online shoppers is less than 50%.
Sample Size (n): 850
Significance Level (α): 5%

z=p-PP(1-P)/n

z=0.47-0.500.50.5/850 −1.74

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is -1.645. Since -1.74 is less than -1.645, we reject the null hypothesis. This means the data does not support the idea that most customers are moving towards online grocery shopping.

In a study of code quality, Team A has 250 errors in 1000 lines of code, and Team B has 300 errors in 800 lines of code. Can we say Team B performs worse than Team A?

Solution: Let’s analyze it:

Proportion of errors for Team A (pA): 250 / 1000 = 0.25
Proportion of errors for Team B (pB): 300 / 800 = 0.375
Null Hypothesis (H0): Team B’s error rate is less than or equal to Team A’s.
Alternate Hypothesis (H1): Team B’s error rate is greater than Team A’s.
Sample Size for Team A (nA): 1000
Sample Size for Team B (nB): 800

p=nApA+nBpBnA+nB

p=10000.25+8000.3751000+800 ≈ 0.305

z=pA−pBp(1-p)(1nA+1nB)

z=0.25−0.3750.305(1-0.305) (11000+1800) ≈ −5.72

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is +1.645. Since -5.72 is far less than +1.645, we reject the null hypothesis. The data indicates that Team B’s performance is significantly worse than Team A’s.

Our Data Scientist Master's Program will help you master core topics such as R, Python, Machine Learning, Tableau, Hadoop, and Spark. Get started on your journey today!

Applications of Hypothesis Testing

Apart from the practical problems, let's look at the real-world applications of hypothesis testing across various fields:

Medicine and Healthcare

In medicine, hypothesis testing plays a pivotal role in assessing the success of new treatments. For example, researchers may want to find out if a new exercise regimen improves heart health. By comparing data from patients who followed the program to those who didn’t, they can determine if the exercise significantly improves health outcomes. Such rigorous testing allows medical professionals to rely on proven methods rather than assumptions.

Quality Control and Manufacturing

In manufacturing, ensuring product quality is vital, and hypothesis testing helps maintain those standards. Suppose a beverage company introduces a new bottling process and wants to verify if it reduces contamination. By analyzing samples from the new and old processes, hypothesis testing can reveal whether the new method reduces the risk of contamination. This allows manufacturers to implement improvements that enhance product safety and quality confidently.

Education and Learning

In education and learning, hypothesis testing is a tool to evaluate the impact of innovative teaching techniques. Imagine a situation where teachers introduce project-based learning to boost critical thinking skills. By comparing the performance of students who engaged in project-based learning with those in traditional settings, educators can test their hypothesis. The results can help educators make informed choices about adopting new teaching strategies.

Environmental Science

Hypothesis testing is essential in environmental science for evaluating the effectiveness of conservation measures. For example, scientists might explore whether a new water management strategy improves river health. By collecting and comparing data on water quality before and after the implementation of the strategy, they can determine whether the intervention leads to positive changes. Such findings are crucial for guiding environmental decisions that have long-term impacts.

Marketing and Advertising

In marketing, businesses use hypothesis testing to refine their approaches. For instance, a clothing brand might test if offering limited-time discounts increases customer loyalty. By running campaigns with and without the discount and analyzing the outcomes, they can assess if the strategy boosts customer retention. Data-driven insights from hypothesis testing enable companies to design marketing strategies that resonate with their audience and drive growth.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

Learn All The Tricks Of The BI Trade

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0 and H1 represent the null and alternative hypotheses. The null hypothesis, H0, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

5. What software tools can assist with hypothesis testing?

Several software tools offering distinct features can help with hypothesis testing. R and RStudio are popular for their advanced statistical capabilities. The Python ecosystem, including libraries like SciPy and Statsmodels, also supports hypothesis testing. SAS and SPSS are well-established tools for comprehensive statistical analysis. For basic testing, Excel offers simple built-in functions.

6. How do I interpret the results of a hypothesis test?

Interpreting hypothesis test results involves comparing the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, you can reject the null hypothesis, indicating statistical significance. This suggests that the observed effect is unlikely to have occurred by chance, validating your analysis findings.

7. Why is sample size important in hypothesis testing?

Sample size is crucial in hypothesis testing as it affects the test’s power. A larger sample size increases the likelihood of detecting a true effect, reducing the risk of Type II errors. Conversely, a small sample may lack the statistical power needed to identify differences, potentially leading to inaccurate conclusions.

8. Can hypothesis testing be used for non-numerical data?

Yes, hypothesis testing can be applied to non-numerical data through non-parametric tests. These tests are ideal when data doesn't meet parametric assumptions or when dealing with categorical data. Non-parametric tests, like the Chi-square or Mann-Whitney U test, provide robust methods for analyzing non-numerical data and drawing meaningful conclusions.

9. How do I choose the proper hypothesis test?

Selecting the right hypothesis test depends on several factors: the objective of your analysis, the type of data (numerical or categorical), and the sample size. Consider whether you're comparing means, proportions, or associations, and whether your data follows a normal distribution. The correct choice ensures accurate results tailored to your research question.

Find our PL-300 Microsoft Power BI Certification Training Online Classroom training classes in top cities:

Name	Date	Place
	21 Sep -6 Oct 2024, Weekend batch	Your City
	12 Oct -27 Oct 2024, Weekend batch	Your City
	26 Oct -10 Nov 2024, Weekend batch	Your City

About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

Recommended Resources

Free eBook: Top Programming Languages For A Data Scientist

Normality Test in Minitab: Minitab with Statistics

Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer

PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
Duis aute irure dolor in reprehenderit in voluptate
Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

1.2 - the 7 step process of statistical hypothesis testing.

We will cover the seven steps one by one.

Step 1: State the Null Hypothesis

The null hypothesis can be thought of as the opposite of the "guess" the researchers made. In the example presented in the previous section, the biologist "guesses" plant height will be different for the various fertilizers. So the null hypothesis would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as:

$H_0 \colon \mu_1 = \mu_2 = ⋯ = \mu_T$

for T levels of an experimental treatment.

Step 2: State the Alternative Hypothesis

$H_A \colon \text{ treatment level means not all equal}$

The alternative hypothesis is stated in this way so that if the null is rejected, there are many alternative possibilities.

For example, $\mu_1\ne \mu_2 = ⋯ = \mu_T$ is one possibility, as is $\mu_1=\mu_2\ne\mu_3= ⋯ =\mu_T$. Many people make the mistake of stating the alternative hypothesis as $\mu_1\ne\mu_2\ne⋯\ne\mu_T$ which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. A simple way of thinking about this is that at least one mean is different from all others. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, a possible outcome would be that fertilizer 1 results in plants that are exceptionally tall, but fertilizers 2, 3, and the control group may not differ from one another.

Step 3: Set $\alpha$

If we look at what can happen in a hypothesis test, we can construct the following contingency table:

Decision	In Reality
Decision	$H_0$ is TRUE	$H_0$ is FALSE
Accept $H_0$	correct	Type II Error $\beta$ = probability of Type II Error
Reject $H_0$	Type I Error $\alpha$ = probability of Type I Error	correct

You should be familiar with Type I and Type II errors from your introductory courses. It is important to note that we want to set $\alpha$ before the experiment ( a-priori ) because the Type I error is the more grievous error to make. The typical value of $\alpha$ is 0.05, establishing a 95% confidence level. For this course, we will assume $\alpha$ =0.05, unless stated otherwise.

Step 4: Collect Data

Remember the importance of recognizing whether data is collected through an experimental design or observational study.

Step 5: Calculate a test statistic

For categorical treatment level means, we use an F- statistic, named after R.A. Fisher. We will explore the mechanics of computing the F- statistic beginning in Lesson 2. The F- value we get from the data is labeled $F_{\text{calculated}}$.

Step 6: Construct Acceptance / Rejection regions

As with all other test statistics, a threshold (critical) value of F is established. This F- value can be obtained from statistical tables or software and is referred to as $F_{\text{critical}}$ or $F_\alpha$. As a reminder, this critical value is the minimum value of the test statistic (in this case $F_{\text{calculated}}$) for us to reject the null.

The F- distribution, $F_\alpha$, and the location of acceptance/rejection regions are shown in the graph below:

Step 7: Based on Steps 5 and 6, draw a conclusion about $H_0$

If $F_{\text{calculated}}$ is larger than $F_\alpha$, then you are in the rejection region and you can reject the null hypothesis with $\left(1-\alpha \right)$ level of confidence.

Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting an $F_{\text{calculated}}$ even greater than what you observe assuming the null hypothesis is true. If by chance, the $F_{\text{calculated}} = F_\alpha$, then the p -value would be exactly equal to $\alpha$. With larger $F_{\text{calculated}}$ values, we move further into the rejection region and the p- value becomes less than $\alpha$. So, the decision rule is as follows:

If the p- value obtained from the ANOVA is less than $\alpha$, then reject $H_0$ in favor of $H_A$.

Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

1.
2.
3.
4.
5.
6.
7.
8.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as $H_{0}$. Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as $H_{1}$ or $H_{a}$. For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, $\alpha$ or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$. $\overline{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation and n is the size of the sample.
t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$. s is the sample standard deviation.
$\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}$. $O_{i}$ is the observed value and $E_{i}$ is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

One sample: z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.
Two samples: z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

One sample: t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$.
Two samples: t = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

$H_{0}$: The population parameter is ≤ some value

$H_{1}$: The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

$H_{0}$: The population parameter is ≥ some value

$H_{1}$: The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

$H_{0}$: the population parameter = some value

$H_{1}$: the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
Step 2: Set up the alternative hypothesis.
Step 3: Choose the correct significance level, $\alpha$, and find the critical value.
Step 4: Calculate the correct test statistic (z, t or $\chi$) and p-value.
Step 5: Compare the test statistic with the critical value or compare the p-value with $\alpha$ to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as $H_{0}$: $\mu$ = 100.

Step 2: The alternative hypothesis is given by $H_{1}$: $\mu$ > 100.

Step 3: As this is a one-tailed test, $\alpha$ = 100% - 95% = 5%. This can be used to determine the critical value.

1 - $\alpha$ = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.

$\mu$ = 100, $\overline{x}$ = 112.5, n = 30, $\sigma$ = 15

z = $\frac{112.5-100}{\frac{15}{\sqrt{30}}}$ = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Probability and Statistics
Data Handling

Important Notes on Hypothesis Testing

Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
It involves the setting up of a null hypothesis and an alternate hypothesis.
There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ > 90 $\overline{x}$ = 110, $\mu$ = 90, n = 5, s = 18. $\alpha$ = 0.05 Using the t-distribution table, the critical value is 2.132 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. $H_{0}$: $\mu$ = 80, $H_{1}$: $\mu$ ≠ 80 $\overline{x}$ = 88, $\mu$ = 80, n = 36, $\sigma$ = 10. $\alpha$ = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ z = $\frac{88-80}{\frac{10}{\sqrt{36}}}$ = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ < 90 $\overline{x}$ = 110, $\mu$ = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = $\frac{82-90}{\frac{18}{\sqrt{6}}}$ t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

go to slide go to slide go to slide

Book a Free Trial Class

FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ and for two samples is z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide $\alpha$ by 2. This is done as there are two rejection regions in the curve.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5.6 Hypothesis Tests in Depth

Establishing the parameter of interest, type of distribution to use, the test statistic, and p -value can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when interpreting the results.

Rare Events

Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. Remember that your assumption is just an assumption; it is not a fact, and it may or may not be true. But your sample data are real and are showing you a fact that seems to contradict your assumption.

$\frac{1}{200}$

Errors in Hypothesis Tests

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:

Figure 5.14: Type I and type II errors
	IS ACTUALLY
Action
	Correct outcome	Type II error
	Type I error	Correct outcome

The four possible outcomes in the table are:

The decision is not to reject H 0 when H 0 is true (correct decision).
The decision is to reject H 0 when H 0 is true (incorrect decision known as a type I error ).
The decision is not to reject H 0 when, in fact, H 0 is false (incorrect decision known as a type II error ).
The decision is to reject H 0 when H 0 is false (correct decision whose probability is called the power of the test).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a type I error = P (type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. These are also known as false positives. We know that α is often determined in advance, and α = 0.05 is often widely accepted. In that case, you are saying, “We are OK making this type of error in 5% of samples.” In fact, the p -value is the exact probability of a type I error based on what you observed.

β = probability of a type II error = P (type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. These are also known as false negatives.

The power of a test is 1 – β .

Ideally, α and β should be as small as possible because they are probabilities of errors but are rarely zero. We want a high power that is as close to one as well. Increasing the sample size can help us achieve these by reducing both α and β and therefore increasing the power of the test.

Suppose the null hypothesis, H 0 , is that Frank’s rock climbing equipment is safe.

Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.

α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the type II error, in which Frank thinks his rock climbing equipment is safe, so he goes ahead and uses it.

Suppose the null hypothesis, H 0 , is that the blood cultures contain no traces of pathogen X . State the type I and type II errors.

Statistical Significance vs. Practical Significance

When the sample size becomes larger, point estimates become more precise and any real differences in the mean and null value become easier to detect and recognize. Even a very small difference would likely be detected if we took a large enough sample. Sometimes, researchers will take such large samples that even the slightest difference is detected, even differences where there is no practical value. In such cases, we still say the difference is statistically significant , but it is not practically significant.

For example, an online experiment might identify that placing additional ads on a movie review website statistically significantly increases viewership of a TV show by 0.001%, but this increase might not have any practical value.

One role of a data scientist in conducting a study often includes planning the size of the study. The data scientist might first consult experts or scientific literature to learn what would be the smallest meaningful difference from the null value. She also would obtain other information, such as a very rough estimate of the true proportion p , so that she could roughly estimate the standard error. From here, she could suggest a sample size that is sufficiently large enough to detect the real difference if it is meaningful. While larger sample sizes may still be used, these calculations are especially helpful when considering costs or potential risks, such as possible health impacts to volunteers in a medical study.

Click here for more multimedia resources, including podcasts, videos, lecture notes, and worked examples.

The decision is to reject the null hypothesis when, in fact, the null hypothesis is true

Erroneously rejecting a true null hypothesis or erroneously failing to reject a false null hypothesis

The probability of failing to reject a true hypothesis

Finding sufficient evidence that the observed effect is not just due to variability, often from rejecting the null hypothesis

Significant Statistics Copyright © 2024 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

What is Hypothesis Testing?
Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
Example : Testing a new drug.
Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

Symbolized by the Greek letter alpha (α).
Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

Symbolized by the Greek letter beta (β).
Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We first choose a significance level ($α$), which sets a threshold for making decisions.

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

Set up Hypotheses : Before starting, you make a prediction:
Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.
Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

The average healing time in the Drug Group is 2 hours.
The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

F statistic formula – explained, correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

Subscribe to Machine Learning Plus for high value data science content

Machine Learning A-Z™: Hands-On Python & R In Data Science

Free sample videos:.

Skip to secondary menu
Skip to main content
Skip to primary sidebar

Statistics By Jim

Making statistics intuitive

Test Statistic: Definition, Types & Formulas

By Jim Frost 10 Comments

What is a Test Statistic?

A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger differences between your sample data and the null hypothesis.

When your test statistic indicates a sufficiently large incompatibility with the null hypothesis, you can reject the null and state that your results are statistically significant—your data support the notion that the sample effect exists in the population . To use a test statistic to evaluate statistical significance, you either compare it to a critical value or use it to calculate the p-value .

Statisticians named the hypothesis tests after the test statistics because they’re the quantity that the tests actually evaluate. For example, t-tests assess t-values, F-tests evaluate F-values, and chi-square tests use, you guessed it, chi-square values.

In this post, learn about test statistics, how to calculate them, interpret them, and evaluate statistical significance using the critical value and p-value methods.

How to Find Test Statistics

Each test statistic has its own formula. I present several common test statistics examples below. To see worked examples for each one, click the links to my more detailed articles.

Formulas for Test Statistics


T-value for 1-sample t-test		Take the sample mean, subtract the hypothesized mean, and divide by the .
T-value for 2-sample t-test		Take one sample mean, subtract the other, and divide by the pooled standard deviation.
F-value for F-tests and ANOVA		Calculate the ratio of two .
Chi-squared value (χ ) for a Chi-squared test		Sum the squared differences between observed and expected values divided by the expected values.

Understanding the Null Values and the Test Statistic Formulas

In the formulas above, it’s helpful to understand the null condition and the test statistic value that occurs when your sample data match that condition exactly. Also, it’s worthwhile knowing what causes the test statistics to move further away from the null value, potentially becoming significant. Test statistics are statistically significant when they exceed a critical value.

All these test statistics are ratios, which helps you understand their null values.

T-Tests, Null = 0

When a t-value equals 0, it indicates that your sample data match the null hypothesis exactly.

For a 1-sample t-test, when the sample mean equals the hypothesized mean, the numerator is zero, which causes the entire t-value ratio to equal zero. As the sample mean moves away from the hypothesized mean in either the positive or negative direction, the test statistic moves away from zero in the same direction.

A similar case exists for 2-sample t-tests. When the two sample means are equal, the numerator is zero, and the entire test statistic ratio is zero. As the two sample means become increasingly different, the absolute value of the numerator increases, and the t-value becomes more positive or negative.

F-tests including ANOVA, Null = 1

When an F-value equals 1, it indicates that the two variances in the numerator and denominator are equal, matching the null hypothesis.

As the numerator and denominator become less and less similar, the F-value moves away from one in either direction.

Chi-squared Tests, Null = 0

When a chi-squared value equals 0, it indicates that the observed values always match the expected values. This condition causes the numerator to equal zero, making the chi-squared value equal zero.

As the observed values progressively fail to match the expected values, the numerator increases, causing the test statistic to rise from zero.

Related post : How a Chi-Squared Test Works

You’ll never see a test statistic that equals the null value precisely in practice. However, trivial differences been sample values and the null value are not uncommon.

Interpreting Test Statistics

Test statistics are unitless. This fact can make them difficult to interpret on their own. You know they evaluate how well your data agree with the null hypothesis. If your test statistic is extreme enough, your data are so incompatible with the null hypothesis that you can reject it and conclude that your results are statistically significant. But how does that translate to specific values of your test statistic? Where do you draw the line?

For instance, t-values of zero match the null value. But how far from zero should your t-value be to be statistically significant? Is 1 enough? 2? 3? If your t-value is 2, what does it mean anyway? In this case, we know that the sample mean doesn’t equal the null value, but how exceptional is it? To complicate matters, the dividing line changes depending on your sample size and other study design issues.

Similar types of questions apply to the other test statistics too.

To interpret individual values of a test statistic, we need to place them in a larger context. Towards this end, let me introduce you to sampling distributions for test statistics!

Sampling Distributions for Test Statistics

Performing a hypothesis test on a sample produces a single test statistic. Now, imagine you carry out the following process:

Assume the null hypothesis is true in the population.
Repeat your study many times by drawing many random samples of the same size from this population.
Perform the same hypothesis test on all these samples and save the test statistics.
Plot the distribution of the test statistics.

This process produces the distribution of test statistic values that occurs when the effect does not exist in the population (i.e., the null hypothesis is true). Statisticians refer to this type of distribution as a sampling distribution, a kind of probability distribution.

Why would we need this type of distribution?

It provides the larger context required for interpreting a test statistic. More specifically, it allows us to compare our study’s single test statistic to values likely to occur when the null is true. We can quantify our sample statistic’s rareness while assuming the effect does not exist in the population. Now that’s helpful!

Fortunately, we don’t need to collect many random samples to create this distribution! Statisticians have developed formulas allowing us to estimate sampling distributions for test statistics using the sample data.

To evaluate your data’s compatibility with the null hypothesis, place your study’s test statistic in the distribution.

Related post : Understanding Probability Distributions

Example of a Test Statistic in a Sampling Distribution

Suppose our t-test produces a t-value of two. That’s our test statistic. Let’s see where it fits in.

The sampling distribution below shows a t-distribution with 20 degrees of freedom, equating to a 1-sample t-test with a sample size of 21. The distribution centers on zero because it assumes the null hypothesis is correct. When the null is true, your analysis is most likely to obtain a t-value near zero and less likely to produce t-values further from zero in either direction.

Sampling distribution for the t-value test statistic.

The sampling distribution indicates that our test statistic is somewhat rare when we assume the null hypothesis is correct. However, the chances of observing t-values from -2 to +2 are not totally inconceivable. We need a way to quantify the likelihood.

From this point, we need to use the sampling distributions’ ability to calculate probabilities for test statistics.

Related post : Sampling Distributions Explained

Test Statistics and Critical Values

The significance level uses critical values to define how far the test statistic must be from the null value to reject the null hypothesis. When the test statistic exceeds a critical value, the results are statistically significant.

The percentage of the area beneath the sampling distribution curve that is shaded represents the probability that the test statistic will fall in those regions when the null is true. Consequently, to depict a significance level of 0.05, I’ll shade 5% of the sampling distribution furthest away from the null value.

The two shaded areas are equidistant from the null value in the center. Each region has a likelihood of 0.025, which sums to our significance level of 0.05. These shaded areas are the critical regions for a two-tailed hypothesis test. Let’s return to our example t-value of 2.

Related post : What are Critical Values?

Sampling distribution that displays the critical values for our t-value.

In this example, the critical values are -2.086 and +2.086. Our test statistic of 2 is not statistically significant because it does not exceed the critical value.

Other hypothesis tests have their own test statistics and sampling distributions, but their processes for critical values are generally similar.

Learn how to find critical values for test statistics using tables:

T-distribution table
Chi-square table

Related post : Understanding Significance Levels

Using Test Statistics to Find P-values

P-values are the probability of observing an effect at least as extreme as your sample’s effect if you assume no effect exists in the population.

Test statistics represent effect sizes in hypothesis tests because they denote the difference between your sample effect and no effect —the null hypothesis. Consequently, you use the test statistic to calculate the p-value for your hypothesis test.

The above p-value definition is a bit tortuous. Fortunately, it’s much easier to understand how test statistics and p-values work together using a sampling distribution graph.

Let’s use our hypothetical test statistic t-value of 2 for this example. However, because I’m displaying the results of a two-tailed test, I need to use t-values of +2 and -2 to cover both tails.

Related post : One-tailed vs. Two-Tailed Hypothesis Tests

The graph below displays the probability of t-values less than -2 and greater than +2 using the area under the curve. This graph is specific to our t-test design (1-sample t-test with N = 21).

Graph of t-distribution that displays the probability for a t-value of 2.

The sampling distribution indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. That’s the p-value! The graph shows that the test statistic falls within these areas almost 6% of the time when the null hypothesis is true in the population.

While this likelihood seems small, it’s not low enough to justify rejecting the null under the standard significance level of 0.05. P-value results are always consistent with the critical value method. Learn more about using test statistics to find p values .

While test statistics are a crucial part of hypothesis testing, you’ll probably let your statistical software calculate the p-value for the test. However, understanding test statistics will boost your comprehension of what a hypothesis test actually assesses.

Reader Interactions

July 5, 2024 at 8:21 am

“As the observed values progressively fail to match the observed values, the numerator increases, causing the test statistic to rise from zero”.

Sir, this sentence is written in the Chi-squared Test heading. There the observed value is written twice. I think the second one to be replaced with ‘expected values’.

July 5, 2024 at 4:10 pm

Thanks so much, Dr. Raj. You’re correct about the typo and I’ve made the correction.

May 9, 2024 at 1:40 am

Thank you very much (great page on one and two-tailed tests)!

May 6, 2024 at 12:17 pm

I would like to ask a question. If only positive numbers are the possible values in a sample (e.g. absolute values without 0), is it meaningful to test if the sample is significantly different from zero (using for example a one sample t-test or a Wilcoxon signed-rank test) or can I assume that if given a large enough sample, the result will by definition be significant (even if a small or very variable sample results in a non-significant hypothesis test).

Thank you very much,

May 6, 2024 at 4:35 pm

If you’re talking about the raw values you’re assessing using a one-sample t-test, it doesn’t make sense to compare them to zero given your description of the data. You know that the mean can’t possibly equal zero. The mean must be some positive value. Yes, in this scenario, if you have a large enough sample size, you should get statistically significant results. So, that t-test isn’t tell you anything that you don’t already know!

However, you should be aware of several things. The 1-sample test can compare your sample mean to values other than zero. Typically, you’ll need to specify the value of the null hypothesis for your software. This value is the comparison value. The test determines whether your sample data provide enough evidence to conclude that the population mean does not equal the null hypothesis value you specify. You’ll need to specify the value because there is no obvious default value to use. Every 1-sample t-test has its subject-area context with a value that makes sense for its null hypothesis value and it is frequently not zero.

I suspect that you’re getting tripped up with the fact that t-tests use a t-value of zero for its null hypothesis value. That doesn’t mean your 1-sample t-test is comparing your sample mean to zero. The test converts your data to a single t-value and compares the t-value to zero. But your actual null hypothesis value can be something else. It’s just converting your sample to a standardized value to use for testing. So, while the t-test compares your sample’s t-value to zero, you can actually compare your sample mean to any value you specify. You need to use a value that makes sense for your subject area.

I hope that makes sense!

May 8, 2024 at 8:37 am

Thank you very much Jim, this helps a lot! Actually, the value I would like to compare my sample to is zero, but I just couldn’t find the right way to test it apparently (it’s about EEG data). The original data was a sample of numbers between -1 and +1, with the question if they are significantly different from zero in either direction (in which case a one sample t-test makes sense I guess, since the sample mean can in fact be zero). However, since a sample mean of 0 can also occur if half of the sample differs in the negative, and the other half in the positive direction, I also wanted to test if there is a divergence from 0 in ‘absolute’ terms – that’s how the absolute valued numbers came about (I know that absolute values can also be zero, but in this specific case, they were all positive numbers) And a special thanks for the last paragraph – I will definitely keep in mind, it is a potential point of confusion.

May 8, 2024 at 8:33 pm

You can use a 1-sample t test for both cases but you’ll need to set them up slightly different. To detect a positive or negative difference from zero, use a 2-tailed test. For the case with absolute values, use a one-tailed test with a critical region in the positive end. To learn more, read about One- and Two-Tailed Tests Explained . Use zero for the comparison value in both cases.

February 12, 2024 at 1:00 am

Very helpful and well articulated! Thanks Jim 🙂

September 18, 2023 at 10:01 am

Thank you for brief explanation.

July 25, 2022 at 8:32 am

the content was helpful to me. thank you

Comments and Questions Cancel reply

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

Knowledge Base

Test statistics | Definition, Interpretation, and Examples

Published on July 17, 2020 by Rebecca Bevans . Revised on June 22, 2023.

The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test.

The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.

What exactly is a test statistic, types of test statistics, interpreting test statistics, reporting test statistics, other interesting articles, frequently asked questions about test statistics.

A test statistic describes how closely the distribution of your data matches the distribution predicted under the null hypothesis of the statistical test you are using.

The distribution of data is how often each observation occurs, and can be described by its central tendency and variation around that central tendency. Different statistical tests predict different types of distributions, so it’s important to choose the right statistical test for your hypothesis.

The test statistic summarizes your observed data into a single number using the central tendency, variation, sample size, and number of predictor variables in your statistical model.

Generally, the test statistic is calculated as the pattern in your data (i.e., the correlation between variables or difference between groups) divided by the variance in the data (i.e., the standard deviation ).

Null hypothesis ( H 0 ): There is no correlation between temperature and flowering date.
Alternate hypothesis ( H A or H 1 ): There is a correlation between temperature and flowering date.

Prevent plagiarism. Run a free check.

Below is a summary of the most common test statistics, their hypotheses, and the types of statistical tests that use them.

Different statistical tests will have slightly different ways of calculating these test statistics, but the underlying hypotheses and interpretations of the test statistic stay the same.

Test statistic	Null and alternative hypotheses	Statistical tests that use it
value	The means of two groups are equal The means of two groups are not equal	test
value	The means of two groups are equal The means of two groups are not equal	test
value	The variation among two or more groups is greater than or equal to the variation between the groups The variation among two or more groups is smaller than the variation between the groups
-value	Two samples are independent Two samples are not independent (i.e., they are correlated)	correlation tests

In practice, you will almost always calculate your test statistic using a statistical program (R, SPSS, Excel, etc.), which will also calculate the p value of the test statistic. However, formulas to calculate these statistics by hand can be found online.

a regression coefficient of 0.36
a t value comparing that coefficient to the predicted range of regression coefficients under the null hypothesis of no relationship

The t value of the regression test is 2.36 – this is your test statistic.

For any combination of sample sizes and number of predictor variables, a statistical test will produce a predicted distribution for the test statistic. This shows the most likely range of values that will occur if your data follows the null hypothesis of the statistical test.

The more extreme your test statistic – the further to the edge of the range of predicted test values it is – the less likely it is that your data could have been generated under the null hypothesis of that statistical test.

The agreement between your calculated test statistic and the predicted values is described by the p value . The smaller the p value, the less likely your test statistic is to have occurred under the null hypothesis of the statistical test.

Because the test statistic is generated from your observed data, this ultimately means that the smaller the p value, the less likely it is that your data could have occurred if the null hypothesis was true.

Test statistics can be reported in the results section of your research paper along with the sample size, p value of the test, and any characteristics of your data that will help to put these results into context.

Whether or not you need to report the test statistic depends on the type of test you are reporting.

	Which statistics to report
Correlation and regression tests	or regression coefficient for each predictor variable value for each predictor
Tests of difference between groups	value for the test statistic

By surveying a random subset of 100 trees over 25 years we found a statistically significant ( p < 0.01) positive correlation between temperature and flowering dates ( R 2 = 0.36, SD = 0.057).

In our comparison of mouse diet A and mouse diet B, we found that the lifespan on diet A ( M = 2.1 years; SD = 0.12) was significantly shorter than the lifespan on diet B ( M = 2.6 years; SD = 0.1), with an average difference of 6 months ( t (80) = -12.75; p < 0.01).

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

Academic style
Vague sentences
Style consistency

See an example

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

Confidence interval
Descriptive statistics
Measures of central tendency
Correlation coefficient

Methodology

Cluster sampling
Stratified sampling
Types of interviews
Cohort study
Thematic analysis

Research bias

Implicit bias
Cognitive bias
Survivorship bias
Availability heuristic
Nonresponse bias
Regression to the mean

A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).

The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bevans, R. (2023, June 22). Test statistics | Definition, Interpretation, and Examples. Scribbr. Retrieved September 9, 2024, from https://www.scribbr.com/statistics/test-statistic/

Is this article helpful?

Rebecca Bevans

Other students also liked, understanding p values | definition and examples, choosing the right statistical test | types & examples, what is effect size and why does it matter (examples), what is your plagiarism score.

Warning: The NCBI web site requires JavaScript to function. more...

An official website of the United States government

The .gov means it's official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you're on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Publications
Account settings
Browse Titles

NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health.

StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

Affiliations

Last Update: March 13, 2023 .

Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

Review Questions
Access free multiple choice questions on this topic.
Comment on this article.

Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

In this Page

Bulk download.

Bulk download StatPearls data from FTP

Related information

PMC PubMed Central citations
PubMed Links to PubMed

Recent Activity

Hypothesis Testing, P Values, Confidence Intervals, and Significance - StatPearl... Hypothesis Testing, P Values, Confidence Intervals, and Significance - StatPearls

Your browsing activity is empty.

Activity recording is turned off.

Turn recording back on

Connect with NLM

National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894

Web Policies FOIA HHS Vulnerability Disclosure

Help Accessibility Careers

LEARN STATISTICS EASILY

Learn Data Analysis Now!

A Comprehensive Guide to Hypotheses Tests in Statistics

You will learn the essentials of hypothesis tests, from fundamental concepts to practical applications in statistics.

Null and alternative hypotheses guide hypothesis tests.
Significance level and p-value aid decision-making.
Parametric tests assume specific probability distributions.
Non-parametric tests offer flexible assumptions.
Confidence intervals provide estimate precision.

Introduction to Hypotheses Tests

Hypothesis testing is a statistical tool used to make decisions based on data.

It involves making assumptions about a population parameter and testing its validity using a population sample.

Hypothesis tests help us draw conclusions and make informed decisions in various fields like business, research, and science.

Null and Alternative Hypotheses

The null hypothesis (H0) is an initial claim about a population parameter, typically representing no effect or no difference.

The alternative hypothesis (H1) opposes the null hypothesis, suggesting an effect or difference.

Hypothesis tests aim to determine if there is evidence for the null hypothesis rejection in favor of the alternative hypothesis.

Significance Levels and P-values

The significance level (α), often set at 0.05 or 5%, serves as a threshold for determining if we should reject the null hypothesis.

A p-value, calculated during hypothesis testing, represents the probability of observing the test statistic if the null hypothesis is true.

Suppose the p-value is less than the significance level. We reject the null hypothesis, in that case, indicating that the alternative hypothesis is more likely.

Parametric and Non-Parametric Tests

Parametric tests assume the data follows a specific probability distribution, usually the normal distribution. Examples include the Student’s t-test.

Non-parametric tests do not require such assumptions and are helpful when dealing with data that do not meet the assumptions of parametric tests. Examples include the Mann-Whitney U test.

🎓 Master Data Analysis and Skyrocket Your Career

Find Out the Secrets in Our Ultimate Guide! 💼

Commonly Used Hypotheses Tests

Independent samples t-test: This analysis compares the means of two independent groups.

Paired samples t-test: Compares the means of two related groups (e.g., before and after treatment).

Chi-squared test: Determines if there is a significant association, in a contingency table, between two categorical variables.

Analysis of Variance (ANOVA): Compares the means of three or more independent groups to determine whether significant differences exist.

Pearson’s Correlation Coefficient (Pearson’s r): Quantifies the strength and direction of a linear association between two continuous variables.

Simple Linear Regression: Evaluate whether a significant linear relationship exists between a predictor variable (X) and a continuous outcome variable (y).

Logistic Regression: Determines the relationship between one or more predictor variables (continuous or categorical) and a binary outcome variable (e.g., success or failure).

Levene’s Test: Tests the equality of variances between two or more groups, often used as an assumption checks for ANOVA.

Shapiro-Wilk Test: Assesses the null hypothesis that a data sample is drawn from a population with a normal distribution.

Hypothesis Test	Description	Application
	Compares means of two independent groups	Comparing scores of two groups of students
	Compares means of two related groups (e.g., before and after treatment)	Comparing weight loss before and after a diet program
	Determines significant associations between two categorical variables in a contingency table	Analyzing the relationship between education and income
	Compares means of three or more independent groups	Evaluating the impact of different teaching methods on test scores
	Measures the strength and direction of a linear relationship between two continuous variables	Studying the correlation between height and weight
	Determines a significant linear relationship between a predictor variable and an outcome variable	Predicting sales based on advertising budget
	Determines the relationship between predictor variables and a binary outcome variable	Predicting the probability of loan default based on credit score
	Tests the equality of variances between two or more groups	Checking the assumption of equal variances for ANOVA
	Tests if a data sample is from a normally distributed population	Assessing normality assumption for parametric tests

Interpreting the Results of Hypotheses Tests

To interpret the hypothesis test results, compare the p-value to the chosen significance level.

If the p-value falls below the significance level, reject the null hypothesis and infer that a notable effect or difference exists.

Otherwise, fail to reject the null hypothesis, meaning there is insufficient evidence to support the alternative hypothesis.

Other Relevant Information

In addition to understanding the basics of hypothesis tests, it’s crucial to consider other relevant information when interpreting the results.

For example, factors such as effect size, statistical power, and confidence intervals can provide valuable insights and help you make more informed decisions.

Effect size

The effect size represents a quantitative measurement of the strength or magnitude of the observed relationship or effect between variables. It aids in evaluating the practical significance of the results. A statistically significant outcome may not necessarily imply practical relevance. At the same time, a substantial effect size can suggest meaningful findings, even when statistical significance appears marginal.

Statistical power

The power of a test represents the likelihood of accurately rejecting the null hypothesis when it is incorrect. In other words, it’s the likelihood that the test will detect an effect when it exists. Factors affecting the power of a test include the sample size, effect size, and significance level. Enhanced power reduces the likelihood of making an error of Type II — failing to reject the null hypothesis when it ought to be rejected.

Confidence intervals

A confidence interval represents a range where the true population parameter is expected to be found with a specified confidence level (e.g., 95%). Confidence intervals provide additional context to hypothesis testing, helping to assess the estimate’s precision and offering a better understanding of the uncertainty surrounding the results.

By considering these additional aspects when interpreting the results of hypothesis tests, you can gain a more comprehensive understanding of the data and make more informed conclusions.

Hypothesis testing is an indispensable statistical tool for drawing meaningful inferences and making informed data-based decisions.

By comprehending the essential concepts such as null and alternative hypotheses, significance levels, p-values, and the distinction between parametric and non-parametric tests, you can proficiently apply hypothesis testing to a wide range of real-world situations.

Additionally, understanding the importance of effect sizes, statistical power, and confidence intervals will enhance your ability to interpret the results and make better decisions.

With many applications across various fields, including medicine, psychology, business, and environmental sciences, hypothesis testing is a versatile and valuable method for research and data analysis.

A comprehensive grasp of hypothesis testing techniques will enable professionals and researchers to strengthen their decision-making processes, optimize strategies, and deepen their understanding of the relationships between variables, leading to more impactful results and discoveries.

Refine your data analysis skills and present meaningful insights with confidence using our latest digital book!

Access FREE samples now and master advanced techniques in data analysis, including optimal sample size determination and effective communication of results.

Don’t miss the chance to immerse yourself in Applied Statistics: Data Analysis and unlock your full potential in data-driven decision making.

Click the link to start exploring!

Can Standard Deviations Be Negative?

Connect with us on our social networks.

DAILY POSTS ON INSTAGRAM!

Hypothesis Tests

Avoiding the Pitfall of Jumping to Conclusions in Data Science

Avoid jumping to conclusions in data science with these statistical insights and methods. Learn how to ensure accuracy and validity.

What Are The 3 Types of Logistic Regression?

Discover the three types of logistic regression: Binary, Ordinal, and Multinomial. Understand their unique applications in statistical analysis and data science.

APA Style T-Test Reporting Guide

How to report t-test in APA style? Learn now! Tips, examples, and effect size calculation included. Learn how to report t-test results.

Understanding Data: Why ‘Statistics Are Like Bikinis’

Statistics are like bikinis: revealing crucial data insights and acknowledging hidden aspects for a balanced data interpretation approach.

Understanding Overconfidence in Statistics: Quantifying Uncertainty Accurately

Discover the impact of overconfidence in statistics and learn how to quantify uncertainty using statistical methods accurately.

Which Normality Test Should You Use?

Discover the best normality test for data analysis based on Razali and Wah’s study comparing 4 popular tests. Read now!

What Is Hypothesis Testing?

How It Works

4 Step Process

The bottom line.

Fundamental Analysis

Hypothesis Testing: 4 Steps and Example

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
The test provides evidence concerning the plausibility of the hypothesis, given the data.
Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

State the hypotheses.
Formulate an analysis plan, which outlines how the data will be evaluated.
Carry out the plan and analyze the sample data.
Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

When Did Hypothesis Testing Begin?

Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

Terms of Service
Editorial Policy
Privacy Policy

What Is Hypothesis Testing? An In-Depth Guide with Python Examples

Hypothesis testing allows us to make data-driven decisions by testing assertions about populations. It is the backbone behind scientific research, business analytics, financial modeling, and more.

This comprehensive guide aims to solidify your understanding with:

Explanations of key terminology and the overall hypothesis testing process
Python code examples for t-tests, z-tests, chi-squared, and other methods
Real-world examples spanning science, business, politics, and technology
A frank discussion around limitations and misapplications
Next steps to mastering practical statistics with Python

So let‘s get comfortable with making statements, gathering evidence, and letting the data speak!

Fundamentals of Hypothesis Testing

Hypothesis testing is structured around making a claim in the form of competing hypotheses, gathering data, performing statistical tests, and making decisions about which hypothesis the evidence supports.

Here are some key terms about hypotheses and the testing process:

Null Hypothesis ($H_0$): The default statement about a population parameter. Generally asserts that there is no statistical significance between two data sets or that a sample parameter equals some claimed population parameter value. The statement being tested that is either rejected or supported.

Alternative Hypothesis ($H_1$): The statement that sample observations indicate statistically significant effect or difference from what the null hypothesis states. $H_1$ and $H_0$ are mutually exclusive, meaning if statistical tests support rejecting $H_0$, then you conclude $H_1$ has strong evidence.

Significance Level ($\alpha$): The probability of incorrectly rejecting a true null hypothesis, known as making a Type I error. Common significance levels are 90%, 95%, and 99%. The lower significance level, the more strict the criteria is for rejecting $H_0$.

Test Statistic: Summary calculations of sample data including mean, proportion, correlation coefficient, etc. Used to determine statistical significance and improbability under $H_0$.

P-value: Probability of obtaining sample results at least as extreme as the test statistic, assuming $H_0$ is true. Small p-values indicate strong statistical evidence against the null hypothesis.

Type I Error: Incorrectly rejecting a true null hypothesis

Type II Error : Failing to reject a false null hypothesis

These terms set the stage for the overall process:

1. Make Hypotheses

Define the null ($H_0$) and alternative hypothesis ($H_1$).

2. Set Significance Level

Typical significance levels are 90%, 95%, and 99%. Higher significance means more strict burden of proof for rejecting $H_0$.

3. Collect Data

Gather sample and population data related to the hypotheses under examination.

4. Determine Test Statistic

Calculate relevant test statistics like p-value, z-score, t-statistic, etc along with degrees of freedom.

5. Compare to Significance Level

If the test statistic falls in the critical region based on the significance, reject $H_0$, otherwise fail to reject $H_0$.

6. Draw Conclusions

Make determinations about hypotheses given the statistical evidence and context of the situation.

Now that you know the process and objectives, let’s apply this to some concrete examples.

Python Examples of Hypothesis Tests

We‘ll demonstrate hypothesis testing using Numpy, Scipy, Pandas and simulated data sets. Specifically, we‘ll conduct and interpret:

Two sample t-tests
Paired t-tests
Chi-squared tests

These represent some of the most widely used methods for determining statistical significance between groups.

We‘ll plot the data distributions to check normality assumptions where applicable. And determine if evidence exists to reject the null hypotheses across several scenarios.

Two Sample T-Test with NumPy

Two sample t-tests determine whether the mean of a numerical variable differs significantly across two independent groups. It assumes observations follow approximate normal distributions within each group, but not that variances are equal.

Let‘s test for differences in reported salaries at hypothetical Company X vs Company Y:

$H_0$ : Average reported salaries are equal at Company X and Company Y

$H_1$ : Average reported salaries differ between Company X and Company Y

First we‘ll simulate salary samples for each company based on random normal distributions, set a 95% confidence level, run the t-test using NumPy, then interpret.

The t-statistic of 9.35 shows the difference between group means is nearly 9.5 standard errors. The very small p-value rejects the idea the salaries are equal across a randomly sampled population of employees.

Since the test returned a p-value lower than the significance level, we reject $H_0$, meaning evidence supports $H_1$ that average reported salaries differ between these hypothetical companies.

Paired T-Test with Pandas

While an independent groups t-test analyzes mean differences between distinct groups, a paired t-test looks for significant effects pre vs post some treatment within the same set of subjects. This helps isolate causal impacts by removing effects from confounding individual differences.

Let‘s analyze Amazon purchase data to determine if spending increases during the holiday months of November and December.

$H_0$ : Average monthly spending is equal pre-holiday and during the holiday season

$H_1$ : Average monthly spending increases during the holiday season

We‘ll import transaction data using Pandas, add seasonal categories, then run and interpret the paired t-test.

Since the p-value is below the 0.05 significance level, we reject $H_0$. The output shows statistically significant evidence at 95% confidence that average spending increases during November-December relative to January-October.

Visualizing the monthly trend helps confirm the spike during the holiday months.

Single Sample Z-Test with NumPy

A single sample z-test allows testing whether a sample mean differs significantly from a population mean. It requires knowing the population standard deviation.

Let‘s test if recently surveyed shoppers differ significantly in their reported ages from the overall customer base:

$H_0$ : Sample mean age equals population mean age of 39

$H_1$ : Sample mean age does not equal population mean of 39

Here the absolute z-score over 2 and p-value under 0.05 indicates statistically significant evidence that recently surveyed shopper ages differ from the overall population parameter.

Chi-Squared Test with SciPy

Chi-squared tests help determine independence between categorical variables. The test statistic measures deviations between observed and expected outcome frequencies across groups to determine magnitude of relationship.

Let‘s test if credit card application approvals are independent across income groups using simulated data:

$H_0$ : Credit card approvals are independent of income level

$H_1$ : Credit approvals and income level are related

Since the p-value is greater than the 0.05 significance level, we fail to reject $H_0$. There is not sufficient statistical evidence to conclude that credit card approval rates differ by income categories.

ANOVA with StatsModels

Analysis of variance (ANOVA) hypothesis tests determine if mean differences exist across more than two groups. ANOVA expands upon t-tests for multiple group comparisons.

Let‘s test if average debt obligations vary depending on highest education level attained.

$H_0$ : Average debt obligations are equal across education levels

$H_1$ : Average debt obligations differ based on education level

We‘ll simulate ordered education and debt data for visualization via box plots and then run ANOVA.

The ANOVA output shows an F-statistic of 91.59 that along with a tiny p-value leads to rejecting $H_0$. We conclude there are statistically significant differences in average debt obligations based on highest degree attained.

The box plots visualize these distributions and means vary across four education attainment groups.

Real World Hypothesis Testing

Hypothesis testing forms the backbone of data-driven decision making across science, research, business, public policy and more by allowing practitioners to draw statistically-validated conclusions.

Here is a sample of hypotheses commonly tested:

Ecommerce sites test if interface updates increase user conversions
Ridesharing platforms analyze if surge pricing reduces wait times
Subscription services assess if free trial length impacts customer retention
Manufacturers test if new production processes improve output yields

Pharmaceuticals

Drug companies test efficacy of developed compounds against placebo groups
Clinical researchers evaluate impacts of interventions on disease factors
Epidemiologists study if particular biomarkers differ in afflicted populations
Software engineers measure if algorithm optimizations improve runtime complexity
Autonomous vehicles assess whether new sensors reduce accident rates
Information security analyzes if software updates decrease vulnerability exploits

Politics & Social Sciences

Pollsters determine if candidate messaging influences voter preference
Sociologists analyze if income immobility changed across generations
Climate scientists examine anthropogenic factors contributing to extreme weather

This represents just a sample of the wide ranging real-world applications. Properly formulated hypotheses, statistical testing methodology, reproducible analysis, and unbiased interpretation helps ensure valid reliable findings.

However, hypothesis testing does still come with some limitations worth addressing.

Limitations and Misapplications

While hypothesis testing empowers huge breakthroughs across disciplines, the methodology does come with some inherent restrictions:

Over-reliance on p-values

P-values help benchmark statistical significance, but should not be over-interpreted. A large p-value does not necessarily mean the null hypothesis is 100% true for the entire population. And small p-values do not directly prove causality as confounding factors always exist.

Significance also does not indicate practical real-world effect size. Statistical power calculations should inform necessary sample sizes to detect desired effects.

Errors from Multiple Tests

Running many hypothesis tests by chance produces some false positives due to randomness. Analysts should account for this by adjusting significance levels, pre-registering testing plans, replicating findings, and relying more on meta-analyses.

Poor Experimental Design

Bad data, biased samples, unspecified variables, and lack of controls can completely undermine results. Findings can only be reasonably extended to populations reflected by the test samples.

Garbage in, garbage out definitely applies to statistical analysis!

Assumption Violations

Most common statistical tests make assumptions about normality, homogeneity of variance, independent samples, underlying variable relationships. Violating these premises invalidates reliability.

Transformations, bootstrapping, or non-parametric methods can help navigate issues for sound methodology.

Lack of Reproducibility

The replication crisis impacting scientific research highlights issues around lack of reproducibility, especially involving human participants and high complexity systems. Randomized controlled experiments with strong statistical power provide much more reliable evidence.

While hypothesis testing methodology is rigorously developed, applying concepts correctly proves challenging even among academics and experts!

Next Level Hypothesis Testing Mastery

We‘ve covered core concepts, Python implementations, real-world use cases, and inherent limitations around hypothesis testing. What should you master next?

Parametric vs Non-parametric

Learn assumptions and application differences between parametric statistics like z-tests and t-tests that assume normal distributions versus non-parametric analogs like Wilcoxon signed-rank tests and Mann-Whitney U tests.

Effect Size and Power

Look beyond just p-values to determine practical effect magnitude using indexes like Cohen‘s D. And ensure appropriate sample sizes to detect effects using prospective power analysis.

Alternatives to NHST

Evaluate Bayesian inference models and likelihood ratios that move beyond binary reject/fail-to-reject null hypothesis outcomes toward more integrated evidence.

Tiered Testing Framework

Construct reusable classes encapsulating data processing, visualizations, assumption checking, and statistical tests for maintainable analysis code.

Big Data Integration

Connect statistical analysis to big data pipelines pulling from databases, data lakes and APIs at scale. Productionize analytics.

I hope this end-to-end look at hypothesis testing methodology, Python programming demonstrations, real-world grounding, inherent restrictions and next level considerations provides a launchpad for practically applying core statistics! Please subscribe using the form below for more data science tutorials.

Dr. Alex Mitchell is a dedicated coding instructor with a deep passion for teaching and a wealth of experience in computer science education. As a university professor, Dr. Mitchell has played a pivotal role in shaping the coding skills of countless students, helping them navigate the intricate world of programming languages and software development.

Beyond the classroom, Dr. Mitchell is an active contributor to the freeCodeCamp community, where he regularly shares his expertise through tutorials, code examples, and practical insights. His teaching repertoire includes a wide range of languages and frameworks, such as Python, JavaScript, Next.js, and React, which he presents in an accessible and engaging manner.

Dr. Mitchell’s approach to teaching blends academic rigor with real-world applications, ensuring that his students not only understand the theory but also how to apply it effectively. His commitment to education and his ability to simplify complex topics have made him a respected figure in both the university and online learning communities.

Intro to Property-Based Testing in Python

Property-based testing is an innovative technique for testing software through specifying invariant properties rather than manual…

What is Docker? Learn How to Use Containers – Explained with Examples

Docker‘s lightweight container virtualization has revolutionized development workflows. This comprehensive guide demystifies Docker fundamentals while equipping…

How to Create a Timeline Component with React

As a full-stack developer, building reusable UI components is a key skill. In this comprehensive 3200+…

A Brief History of the Command Line

The command line interface (CLI) has been a constant companion of programmers, system administrators and power…

Why I love Vim: It’s the lesser-known features that make it so amazing

Credit: Unsplash Vim has been my go-to text editor for years. As a full-stack developer, I…

How I Started the Process of Healing a Dying Software Group

As the new manager of a struggling 20-person software engineering team, I faced serious challenges that…

School Guide
Mathematics
Number System and Arithmetic
Trigonometry
Probability
Mensuration
Maths Formulas
Class 8 Maths Notes
Class 9 Maths Notes
Class 10 Maths Notes
Class 11 Maths Notes
Class 12 Maths Notes

How to Calculate Test Statistic

In statistical hypothesis testing, a test statistic is a crucial tool used to determine the validity of the hypothesis about a population parameter. This article delves into the calculation of test statistics exploring its importance in hypothesis testing and its application in real-world scenarios. Understanding how to compute and interpret test statistics is essential for students and professionals in various fields including data analysis, research and quality control.

Table of Content

Test Statistic

Types of test statistic, z-statistic, t-statistic, chi-square statistic, f-statistic, examples with solutions, example for z-statistic, example for t-statistic, example for chi-square statistic, example for f-statistic.

A test statistic is a value calculated from sample data during a hypothesis test. It is used to decide whether to reject the null hypothesis. The test statistic measures how far the sample data is from what we would expect under the null hypothesis. Depending on the type of test (e.g., t-test, chi-square test, etc.), the test statistic is compared to a critical value or used to calculate a p-value, which helps in determining the statistical significance of the results.

In simpler terms, think of a test statistic as a number that tells us how much the sample data stands out from what we expect if there’s no real effect or difference. If this number is big enough, we might conclude that something interesting is happening in the data.

There are many types of test statistic:

When the sample size is large and population variance is known, we can use z-statistic.

Formula for Z-Statistic is:

[Tex]Z = \frac{\bar{X} – \mu}{\frac{\sigma}{\sqrt{n}}} [/Tex]

[Tex]\bar{X}[/Tex] = Sample mean
[Tex]\mu[/Tex] = Population mean
[Tex]\sigma[/Tex] = Population standard deviation
n = Sample size

Practice Questions

Question 1: A sample of 50 students has an average height of 165 cm. The population standard deviation is 8 cm. Test if the sample mean is significantly different from the 170 cm at a 0.01 significance level.

Question 2: An online retailer claims that 40% of their customers are repeat buyers. A survey of 200 customers shows that 85 are repeat buyers. Test this claim at a 0.05 significance level.

Question 3: A factory claims that the average lifespan of its light bulbs is 1200 hours. A sample of 20 bulbs has an average lifespan of 1180 hours with the standard deviation of the 50 hours. Test the factory’s claim at a 0.05 significance level.

Question 4: A researcher wants to test if there is a significant difference in the mean scores of two different teaching methods. Method A has a mean score of 85 with a standard deviation of 10 and Method B has a mean score of 80 with the standard deviation of 12. Assume the sample size for both the methods is 25. Test the hypothesis at the 0.05 significance level.

Question 5: A company wants to test if their new product’s defect rate is less than 5%. A sample of 150 products shows that 6 are defective. Test the claim at a 0.01 significance level.

Question 6: We have two independent samples with the following the statistics: Sample 1 (n=15, mean=25, variance=9) and Sample 2 (n=20, mean=22, variance=16). Test if the variances are equal at a 0.05 significance level.

Question 7: A drug manufacturer wants to test if the average recovery time with their new drug is less than the historical average of 30 days. A sample of 12 patients has an average recovery time of 28 days with the standard deviation of 4 days. Test the claim at a 0.05 significance level.

Question 8: In a study of customer satisfaction the variance of the satisfaction scores in two different regions is compared. Region 1 has a variance of 25 and Region 2 has a variance of the 36. The Test if the variances are equal at a 0.05 significance level.

Question 9: An agricultural experiment compares the effects of the two fertilizers on the crop yield. The Fertilizer A yields a mean of 50 kg/acre with the standard deviation of 5 kg/acre and Fertilizer B yields a mean of 55 kg/acre with a standard deviation of the 6 kg/acre. If the sample sizes are both 20 test if the mean yields are significantly different at a 0.05 significance level.

Question 10: A company tests whether the average time to assemble a product is different from expected 45 minutes. The sample of 25 assembly times has a mean of the 47 minutes with the standard deviation of 3 minutes. Test the company’s claim at a 0.05 significance level.

FAQs on How to Calculate Test Statistic

What is a test statistic.

A test statistic is a standardized value used to the test a hypothesis about a population parameter.

How do you choose between a Z-test and a T-test?

The Use a Z-test for the large samples or known population variance and a T-test for the small samples or unknown variance.

What is the significance level in hypothesis testing?

The significance level is the probability of the rejecting the null hypothesis when it is actually true commonly set at 0.05 or 0.01.

How do you interpret the results of a chi-square test?

The Compare the chi-square statistic to the critical values from the chi-square distribution to the determine if there is a significant difference between the observed and expected frequencies.

What is the purpose of the F-statistic in ANOVA?

The F-statistic is used to the compare the variances between the different groups to the determine if there are significant differences among group means.

Please Login to comment...

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

This week: the arXiv Accessibility Forum

Help | Advanced Search

Mathematics > Statistics Theory

Title: a continuous generalization of hypothesis testing.

Abstract: Testing has developed into the fundamental statistical framework for falsifying hypotheses. Unfortunately, tests are binary in nature: a test either rejects a hypothesis or not. Such binary decisions do not reflect the reality of many scientific studies, which often aim to present the evidence against a hypothesis and do not necessarily intend to establish a definitive conclusion. To solve this, we propose the continuous generalization of a test, which we use to measure the evidence against a hypothesis. Such a continuous test can be interpreted as a non-randomized interpretation of the classical 'randomized test'. This offers the benefits of a randomized test, without the downsides of external randomization. Another interpretation is as a literal measure, which measures the amount of binary tests that reject the hypothesis. Our work also offers a new perspective on the $e$-value: the $e$-value is recovered as a continuous test with $\alpha \to 0$, or as an unbounded measure of the amount of rejections.

Subjects:	Statistics Theory (math.ST)
Cite as:	[math.ST]
	(or [math.ST] for this version)
	Focus to learn more arXiv-issued DOI via DataCite (pending registration)

Submission history

Access paper:.

HTML (experimental)
Other Formats

References & Citations

Google Scholar
Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Pardon Our Interruption

As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen:

You've disabled JavaScript in your web browser.
You're a power user moving through this website with super-human speed.
You've disabled cookies in your web browser.
A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article .

To regain access, please make sure that cookies and JavaScript are enabled before reloading the page.

IMAGES

Hypothesis Testing Solved Examples(Questions and Solutions)
Hypothesis Testing
Master Hypothesis Testing in Statistics Guide
Statistical Hypothesis Testing: Step by Step
hypothesis test formula statistics
Hypothesis Testing

VIDEO

Hypothsis Testing in Statistics Part 2 Steps to Solving a Problem
Statistics for Hypothesis Testing
Hypothesis Testing 🔥 Explained in 60 Seconds
Hypothesis Testing Made Easy: These are the Steps
Hypothesis test(One sample mean) using Excel|| Ep-21|| ft.Nirmal Bajracharya
What is Hypothesis Testing?

COMMENTS

Hypothesis Testing
Learn how to perform hypothesis testing, a formal procedure for investigating our ideas about the world using statistics. Follow the 5 steps, from stating your null and alternate hypothesis to presenting your findings.
Statistics
Learn how to check if a claim about a population parameter is true or not using hypothesis testing. Compare the critical value and p-value approaches, and the conditions and steps for testing proportions and means.
Hypothesis Testing: Uses, Steps & Example
Learn how to use hypothesis testing to evaluate the validity of new theories and effects in populations. Follow the five steps of significance testing with a practical example of a new educational program.
Introduction to Hypothesis Testing
Learn the basics of hypothesis testing, a formal statistical method to test assumptions about population parameters. Find out the types of hypotheses, tests, errors, and examples.
Statistical hypothesis test
A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a p -value computed from the test statistic. Roughly 100 specialized statistical tests have been defined. [ 1 ][ 2 ]
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis. The null hypothesis is usually denoted H0 while the alternative hypothesis is usually denoted H1. An hypothesis test is a statistical decision; the conclusion will either be ...
Statistical Hypothesis Testing Overview
Learn the basics of hypothesis testing, a form of inferential statistics that uses sample data to assess two mutually exclusive theories about the properties of a population. Find out how to use null and alternative hypotheses, p-values, significance levels, and test statistics to make inferences about a population.
S.3 Hypothesis Testing
Learn the general idea, steps, and errors of hypothesis testing in statistics. Compare the critical value and P-value approaches for testing different types of hypotheses about population parameters.
7.1: Basics of Hypothesis Testing
The null hypothesis has the same parameter and number with an equal sign. H0: μ = $30, 000 HA: μ> $30, 000. b. x = number od students who like math. p = proportion of students who like math. The guess is that p < 0.10 and that is the alternative hypothesis. H0: p = 0.10 HA: p <0.10. c. x = age of students in this class.
Hypothesis Testing
Learn how to perform a hypothesis test, a statistical inference method to test the significance of a proposed relation between population parameters and sample estimators. See definitions, methodology, examples, and confidence intervals.
A Complete Guide to Hypothesis Testing
2. Photo from StepUp Analytics. Hypothesis testing is a method of statistical inference that considers the null hypothesis H ₀ vs. the alternative hypothesis H a, where we are typically looking to assess evidence against H ₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard.
Hypothesis Testing
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
Hypothesis Testing in Statistics
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. ...
1.2
Step 1: State the Null Hypothesis. The null hypothesis can be thought of as the opposite of the "guess" the researchers made. In the example presented in the previous section, the biologist "guesses" plant height will be different for the various fertilizers. So the null hypothesis would be that there will be no difference among the groups of ...
Hypothesis Testing
Learn how to perform hypothesis testing to make statistical inferences about the population data. Find out the types, steps, formulas, and examples of hypothesis testing with z, t, and chi-square tests.
Choosing the Right Statistical Test
What does a statistical test do? Statistical tests work by calculating a test statistic - a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.. It then calculates a p value (probability value). The p-value estimates how likely it is that you would see the difference described by the test statistic if the null ...
5.6 Hypothesis Tests in Depth
5.6 Hypothesis Tests in Depth Establishing the parameter of interest, type of distribution to use, the test statistic, and p-value can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when interpreting the results. ... Statistical Significance vs. Practical Significance. When ...
Hypothesis Testing
Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions. In this Blog post we will learn: What is Hypothesis Testing? Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3.
Test Statistic: Definition, Types & Formulas
Learn how to calculate and interpret test statistics for different hypothesis tests, such as t-tests, F-tests, and chi-squared tests. Test statistics measure how consistent your sample data are with the null hypothesis and help you evaluate statistical significance.
Test statistics
Learn what test statistics are, how they are calculated, and how they help to test hypotheses. Find out the types of test statistics, how to interpret them, and how to report them in your research paper.
Hypothesis Testing, P Values, Confidence Intervals, and Significance
Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting ...
Hypothesis Tests: A Comprehensive Guide
Hypothesis testing is an indispensable statistical tool for drawing meaningful inferences and making informed data-based decisions. By comprehending the essential concepts such as null and alternative hypotheses, significance levels, p-values, and the distinction between parametric and non-parametric tests, you can proficiently apply hypothesis ...
Hypothesis Testing: 4 Steps and Example
Hypothesis testing is the process that an analyst uses to test a statistical hypothesis. The methodology depends on the nature of the data used and the reason for the analysis.
Understanding Hypothesis Testing
Learn the basics of hypothesis testing, a statistical method that evaluates assumptions about population parameters based on sample data. Find out how to formulate null and alternative hypotheses, choose significance level, calculate test statistic, and avoid Type I and II errors.
8.2: The controversy over proper hypothesis testing
This framework over proper hypothesis testing is the basis of the Bayesian vs Frequentist controversy. Consider the independent sample t-test (see Chapter 8.5 and 8.6), our first example of a parametric test. \ ... part of the solution is to consider effect size — introduced in Chapter 9.2 — and the statistical power of the test, see Ch 11 ...
What Is Hypothesis Testing? An In-Depth Guide with Python Examples
Connect statistical analysis to big data pipelines pulling from databases, data lakes and APIs at scale. Productionize analytics. I hope this end-to-end look at hypothesis testing methodology, Python programming demonstrations, real-world grounding, inherent restrictions and next level considerations provides a launchpad for practically applying core statistics!
How to Calculate Test Statistic
In statistical hypothesis testing, a test statistic is a crucial tool used to determine the validity of the hypothesis about a population parameter. This article delves into the calculation of test statistics exploring its importance in hypothesis testing and its application in real-world scenarios. Understanding how to compute and interpret ...
[2409.05654] A Continuous Generalization of Hypothesis Testing
Testing has developed into the fundamental statistical framework for falsifying hypotheses. Unfortunately, tests are binary in nature: a test either rejects a hypothesis or not. Such binary decisions do not reflect the reality of many scientific studies, which often aim to present the evidence against a hypothesis and do not necessarily intend to establish a definitive conclusion. To solve ...
ANOVA, Hypothesis Testing, Descriptive Statistics, Regression
Statistics document from California State University, Fullerton, 2 pages, ISDS 361A - Final Cheat Sheet Problem: Mr. Seinfeld owns three restaurants. He recruited five fellow workers to serve as inspectors and their role is to grade the restaurants on a scale of 0-100. Main Null Hypothesis (number of treatments (k=3): µ(1)=µ(2)

Decision	In Reality
Decision	\(H_0\) is TRUE	\(H_0\) is FALSE
Accept \(H_0\)	correct	Type II Error \(\beta\) = probability of Type II Error
Reject \(H_0\)	Type I Error \(\alpha\) = probability of Type I Error	correct

Statistics Tutorial

Hypothesis Testing

The Null and Alternative Hypothesis

The Significance Level

The Test Statistic

The Critical Value and P-Value Approach

The Critical Value Approach

The P-Value Approach

Steps for a Hypothesis Test

COLOR PICKER

Contact Sales

Report Error

Top Tutorials

Hypothesis Testing

Definitions and Methodology

Hypothesis Testing

What is a Hypothesis?

Bayesian Hypothesis Testing.

What is the Null Hypothesis?

What is a Hypothesis Statement?

How do I State the Null Hypothesis?

Hypothesis Testing Examples #1: Basic Example

Rejecting the null hypothesis

Hypothesis Testing Examples (One Sample Z Test)

One Sample Hypothesis Testing Example: One Tailed Z Test

One Sample Hypothesis Testing Examples: #3

Hypothesis Testing Examples: Mean (Using TI 83)

Bayesian Hypothesis Testing: What is it?

Differences Between Traditional and Bayesian Hypothesis Testing.

Arguments for Bayesian Testing.

Arguments against.

Hypothesis Testing Articles

Tutorial Playlist

The Best Guide to Understand Bayes Theorem

A Complete Guide to Get a Grasp of Time Series Analysis

The Complete Guide to Understand Pearson's Correlation

Table of Contents

What Is Hypothesis Testing in Statistics?

The Ultimate Ticket to Top Data Science Job Roles

Importance of Hypothesis Testing in Data Analysis

Avoiding Misleading Conclusions (Type I and Type II Errors)

Making Smarter Choices

Optimizing Business Tactics

Hypothesis Testing Formula

How Hypothesis Testing Works?

Your Dream Career is Just Around The Corner!

Null Hypothesis and Alternative Hypothesis

Become a Data Scientist with Hands-on Training!

Hypothesis Testing Calculation With Examples

Steps in Hypothesis Testing

Formulate Hypotheses

Choose the Significance Level (α)

Select the Appropriate Test

Collect Data

Calculate the Test Statistic

Determine the p-value

Make a Decision

Report the Results

Perform Post-hoc Analysis (if necessary)

Types of Hypothesis Testing

3. Chi-Square

Hypothesis Testing and Confidence Intervals

Simple and Composite Hypothesis Testing

One-Tailed and Two-Tailed Hypothesis Testing

Become a Data Scientist With Real-World Experience

Right Tailed Hypothesis Testing

Left Tailed Hypothesis Testing

Type 1 and Type 2 Error

Serious About Success? Don't Settle for Less

Practice Problems on Hypothesis Testing

Applications of Hypothesis Testing

Medicine and Healthcare

Quality Control and Manufacturing

Education and Learning

Environmental Science

Marketing and Advertising

Limitations of Hypothesis Testing

Learn All The Tricks Of The BI Trade

1. What is hypothesis testing in statistics with example?

2. What is H0 and H1 in statistics?